Find a natural number greater than one that occurs in the Pascal triangle a) more than three times; b) more than four times.
Prove there are no integer solutions for the equation \(3x^2 + 2 = y^2\).
Prove that the sum of
a) any number of even numbers is even;
b) an even number of odd numbers is even;
c) an odd number of odd numbers is odd.
Prove that the product of
a) two odd numbers is odd;
b) an even number with any integer is even.
During the election for the government of the planet of Liars and
Truth-Tellers, \(12\) candidates each
gave a short speech about themselves.
After everyone had spoken, one alien said: “So far, only one lie has
been told today.”
Then another said: “And now two have been said so far.”
The third said: “And now three lies have been told so far,” and so on —
until the twelfth alien said: “And now twelve lies have been told so
far.”
It turned out that at least one candidate had correctly counted how many
lies had been told before their own statement.
How many lies were said that day in total?
Two people play the following game. Each player in turn rubs out 9 numbers (at his choice) from the sequence \(1, 2, \dots , 100, 101\). After eleven such deletions, 2 numbers will remain. The first player is awarded so many points, as is the difference between these remaining numbers. Prove that the first player can always score at least 55 points, no matter how played the second.
A six-digit phone number is given. How many seven-digit numbers are there from which one can obtain this six-digit number by deleting one digit?
Is it possible to transport 50 stone blocks, whose masses are equal to \(370, 372,\dots, 468\) kg, from a quarry on seven 3-tonne trucks?
There is a counter on the chessboard. Two in turn move the counter to an adjacent on one side cell. It is forbidden to put a counter on a cell, which it has already visited. The one who can not make the next turn loses. Who wins with the right strategy?
The city plan is a rectangle of \(5 \times 10\) cells. On the streets, a one-way traffic system is introduced: it is allowed to go only to the right and upwards. How many different routes lead from the bottom left corner to the upper right?